Once upon a time …

Euler’s problem (1936)

Königsberg’s city (nowadays Kaliningrad) is crossed by Pregel river, which runs around the island of Kneiphof on both sides, and has seven bridges

During a walk, is-it possible to pass on all the bridges of the city once and only once?

• König, D. (1936).Theorie der endlichen und unendlichen Graphen.

König, D. (1990).Theory of finite and infinite graphs. Berlin: Birkhauser

• Berge, C. (1958). Théorie des graphes et ses applications. Paris: Dunod.

English edition, Wiley 1961; Methuen & Co, New York 1962;  Dover, New York 2001.

Russian, Moscow 1961; Spanish, Mexico 1962; Roumanian, Bucharest 1969; Chinese, Shanghai    1963;

Some references …

3 fundamental articles to use similitude analysis in the social representations’ domain

Flament (1962). L’analyse de similitude. Cahiers du Centre de Recherche Opérationnelle, 4, 63-97

Degenne, A. & Vergès, P. (1973). Introduction à l’analyse de similitude. Revue française de Sociologie, 14, 471-512

Flament, C., Degenne, A. & Vergès, P. (1971). Similarity Analysis. Paris: Maison des Sciences de l’Homme.

Some references …

Graphs theoryUseful elements

• A graph G G (V, E)

V = {v1, v2, …, vn} that is n Vertices

E = {e1, e2, …, em } that is m Edges

12

n nm

• A graph G (V, E)

V = {1, 2, 3, 4, 5, 6, 7, 8, 9, } E = {(1, 2), (1, 3), (1, 4), …, (8, 9)}

9 9 1 7236

2 2m

Size of the graph

Graphs theoryUseful elements

G(V,E)– V = {1, 2, 3, 4, 5}– E = {(1,2), (1,3), (1,4), (1,5), (2,3), (2,4), (2,5),

(3,4), (3,5), (4,5)}

G(V,E) – V’ = {1, 2, 3}– E’ = {(1,2), (1,3), (2,3)}

Some vertices = subgraph of G

G(V,E) – V’ = {1, 2, 3, 4, 5}

– E’ = {(1,2), (3,4), (4,5)}

All the vertices, some edges = Spanning Subgraph of G

Graphs theoryUseful elements

G(V,E)– V = {1, 2, 3, 4, 5}– E = {(1,2), (1,3), (1,4), (1,5), (2,3), (2,4), (2,5),

(3,4), (3,5), (4,5)}

Symmetrical relations : (5, 4) = (4, 5)

(1, 3) , (3, 5), (5, 4) = a chain of G

Graphs theoryUseful elements

A complete graph– A cycle– A chain

A TREE

A connected tree without cycle– A chain which connects all the

verticies– A chain without cycle

How to pass from a complete graph to a tree ?

Useful elements for SRHow to pass from a complete graph

to a tree ?

Searching for the structure of the relations =

Searching for the skeleton of the representation=

Searching for a tree

Each edge has a weight=

Similitude analysis ￫ weight = similitude index=

Co-occurrence, symmetrical co-occurrence, Phi square measure,Correlation, squared index of similitude (Guimelli), etc.

Useful elements for SRHow to pass from a complete

graph to a tree ?Searching for the structure of the relations

=Searching for a maximum tree

Searching for a connected graph without cycle +

Searching for the heaviest tree=

Searching for a tree which retains the most similarity

Let us return to our example …

Example 1 (inspired from Abric, 2003)

Stage 4 – From similitude matrix to the structure of the relations between elements

of a representation

finance its leisure activities

the means to earn the keep

personal blooming

self-confidence

the means to have relations

constraints

an obligation

social integration

1

The degree of similitude between two elements can be associated with the graph

5

Arête (7, 6) = (6,7)

Example 1 (inspired from Abric, 2003)

Stage 4 – From similitude matrix to the structure of the relations between elements

of a representation

1

Example 1 (inspired from Abric, 2003)

Stage 4 – From similitude matrix to the structure of the relations between elements

of a representation

1

Edges

S1 Edges

S1 Edges

S1 Edges

S1

(2, 6) 6 (1, 7) 2 (1, 4) 1 (3, 6) 1

(2, 8) 6 (2, 5) 2 (1, 5) 1 (3, 8) 1

(5, 6) 5 (3, 4) 2 (1, 6) 1 (4, 5) 1

(6, 7) 5 (3, 5) 2 (1, 8) 1 (4, 7) 1

(1, 2) 3 (3, 7) 2 (2, 3) 1 (4, 8) 1

(1, 3) 3 (5, 7) 2 (2, 4) 1 (5, 8) 1

(4, 6) 3 (7, 8) 2 (2, 7) 1 (6, 8) 1

Connected graph & without cycle

Important ! (1)

Edges S1 Edges

S1 Edges

S1 Edges

S1

(2, 6) 6 (4, 6) 3 (1, 4) 1 (3, 6) 1

(2, 8) 6 (2, 5) 2 (1, 5) 1 (3, 8) 1

(5, 6) 5 (3, 4) 2 (1, 6) 1 (4, 5) 1

(6, 7) 5 (3, 5) 2 (1, 8) 1 (4, 7) 1

(1, 3) 3 (3, 7) 2 (2, 3) 1 (4, 8) 1

(1, 2) 3 (5, 7) 2 (2, 4) 1 (5, 8) 1

(1, 7) 3 (7, 8) 2 (2, 7) 1 (6, 8) 1

Important ! (2)

Edges S1 Edges S1 Edges S1 Edges S1

(2, 6) 6 (4, 6) 3 (1, 4) 1 (3, 6) 1

(2, 8) 6 (1, 7) 2 (1, 5) 1 (3, 8) 1

(5, 6) 5 (2, 5) 2 (1, 6) 1 (4, 5) 1

(6, 7) 5 (3, 4) 2 (2, 1 ) 1 (4, 7) 1

(7, 8) 5 (3, 5) 2 (2, 3) 1 (4, 8) 1

(1, 8) 3 (3, 7) 2 (2, 4) 1 (5, 8) 1

(1, 3) 3 (5, 7) 2 (2, 7) 1 (6, 8) 1

Example 1 (inspiré de Abric, 2003)

Two populations = 2 graphs

Young students

Workers